Degeneracy in negative feedback (NFBL) and incoherent feedforward (IFFL) loops: Adaptation and resonance

Degeneracy in dynamic models refers to these situations where multiple combinations of parameter values produce identical patterns for the observable variable. We investigate this phenomenon in two qualitatively different adaptive circuit mechanisms: nonlinear feedback loop (NFBL) and incoherent feedback loop (IFFL). We use minimal models of these circuit types together with analytical calculations, regular perturbation analysis, dynamical systems tools and numerical simulations. In response to constant (or step-constant) inputs, NFBLs and IFFLs produce and overshoot allowing the observable variable to return to a value closer to baseline than the peak (adaptation). We identify the dynamic principles underlying the emergence of degeneracy in adaptive patterns both within and across circuit types in representative NFBL and IFFL models in terms of biologically plausible parameters. We identify the conditions under which degeneracy persists in response to oscillatory inputs with arbitrary frequencies, giving rise to resonance and phasonance degeneracy. This naturally extends to the response of adaptive systems to time-dependent inputs within a relatively large class. By using phase-plane analysis, we provide a mechanistic, dynamical systems-based interpretation of degeneracy. Our results have implication for the understanding of adaptive systems, for the relationship between adaptive and resonant/phasonant systems, for the understanding of complex biochemical circuits, for neuronal computation, and for the development of methods for circuit and dynamical systems reconstruction based on experimental or observational data.